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The Mathematical Work of the 2002 Fields Medalists, Volume 50

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The Mathematical Work of the 2002 Fields Medalists, Volume 50 The Mathematical Work of the 2002 Fields Medalists Eric M. Friedlander, Michael Rapoport, and Andrei Suslin The Work of Laurent Lafforgue hand and of the idèle class group A×/F × on the Michael Rapoport other hand. This is the reciprocity law of T. Takagi Laurent Lafforgue was awarded the Fields Medal for and E. Artin established in the 1920s as a far-reach- his proof of the Langlands correspondence for the ing generalization of the quadratic reciprocity law general linear groups GLr over function fields of going back to L. Euler. At the end of the 1960s, positive characteristic. His approach to this prob- R. Langlands proposed a nonabelian generaliza- lem follows the basic strategy introduced twenty- tion of this reciprocity law. It conjecturally relates five years ago by V. Drinfeld in his proof for GL2. the irreducible representations of rank r of Gal(F/F¯ ) Already Drinfeld’s proof is extremely difficult. Laf- (or, more generally, of the hypothetical motivic forgue’s proof is a real tour de force, taking up as Galois group of F) with cuspidal automorphic rep- it does several hundred pages of highly condensed resentations of GL (A). In fact, this conjecture is reasoning. By his achievement Lafforgue has proved r part of an even grander panorama of Langlands (the himself a mathematician of remarkable strength functoriality principle), in which homomorphisms and perseverance. In this brief report I will sketch the background between L-groups of reductive groups over F induce of Lafforgue’s result, state his theorems, and then relations between the automorphic representations mention some ingredients of his proof. The final on the corresponding groups. These hypothetical passages are devoted to the human factor. reciprocity laws would imply famous conjectures, The Background such as the Artin conjecture on the holomorphy of The background of Lafforgue’s theorem is the web L-functions of irreducible Galois representations of conjectures known as the Langlands philosophy, and the Ramanujan-Petersson conjecture on the which is a far-reaching generalization of class field Hecke eigenvalues of cusp forms for GLr. theory. Let F be a global field, i.e., either a finite In the number field case, deep results along extension of Q (the number field case) or a finite ex- these lines have been obtained for groups of small tension of Fp(t) where Fp is the finite field with p rank, such as GL2, by Langlands himself and by elements (the function field case). Let A be the adèle many others. And such results have already had ring of F. spectacular applications, such as in the proof of Global class field theory may be formulated as giving a bijection between the sets of characters of Fermat’s last theorem. However, the proof of the finite order of the Galois group Gal(F/F¯ ) on the one Langlands correspondence in any generality in the number field case seems out of reach at the Michael Rapoport is professor of mathematics at the Uni- present time. Lafforgue’s result, which concerns versität Köln. His email address is rapoport@math. the function field case, is the first general non- uni-koeln.de. abelian reciprocity law. 212 NOTICES OF THE AMS VOLUME 50, NUMBER 2 Lafforgue’s Theorem product formula for ε-factors of G. Laumon), it all From now on let F denote a function field of char- boiled down to proving the existence of the map acteristic p. We also fix an auxiliary prime number π σ (π) with the required properties. This is ex- = p. For a positive integer r let Gr be the set of actly what Lafforgue did. equivalence classes of irreducible -adic repre- Before spending a few words on his proof, let us sentations of dimension r of Gal(F/F¯ ). For each consider the question, What is it good for? The an- σ ∈Gr , A. Grothendieck defined its L-function swer is that neither of the sets Gr and Ar is simpler L(σ,s), which is a rational function in p−s satisfy- than the other in every aspect but that Theorem 1 can ing a functional equation of the form be used to transfer available information in either L(σ,s) = ε(σ,s) · L(σ ∨, 1 − s) , where ε(σ,s) is a direction. Theorem 3 is an instance where informa- −s ∨ monomial in p and σ denotes the contragredi- tion available on Ar implies results on Gr. In the ent representation. The L-function is an Euler prod- other direction, Theorem 1 permits one to use uct, L(σ,s) = ΠxLx(σ,s), over all places x of F, and constructions available on Gr to prove various in- for a place x of degree deg(x), where σ is unrami- stances of Langlands functoriality for Ar, such as fied, we have the existence of tensor products, of base change, r and of automorphic induction. 1 Lx(σ,s) = . About the Proof 1 − z p−s deg(x) i=1 i The strategy of constructing the map π σ (π) is due to Drinfeld and is inspired by the work of Here z1,...,zr are the Frobenius eigenvalues of σ at x. Y. Ihara, Langlands, and others in the theory of Shimura varieties. It consists in analyzing the -adic Let Ar be the set of equivalence classes of cus- pidal representations of GL (A). For each π ∈A, cohomology of the algebraic stack Shtr,∅ over r r × R. Godement and H. Jacquet defined its L-function Spec F Spec F parametrizing shtukas of rank r = L(π,s) with properties similar to those of the above or the algebraic stack Shtr lim← Shtr,N parame- L-functions. The Euler factor at a place x where π trizing shtukas of rank r equipped with a com- is unramified is given as patible system of level structures for all levels N. The latter cohomology module is equipped with an r 1 action of GL (A) × Gal(F/F¯ ) × Gal(F/F¯ ) , and the L (π,s) = , r x − −s deg(x) aim is to isolate inside it a subquotient of the form = 1 zi p i 1 ⊗ ⊗ ∨ where z1,...,zr are the Hecke eigenvalues of π π σ (π) σ (π) at x. The main result of Lafforgue consists of the π∈Ar following theorems. by comparing the Grothendieck-Lefschetz fixed- Theorem 1 (the Langlands conjecture). There is point formula and the Arthur-Selberg trace for- a bijection π σ (π) between Ar and Gr, charac- mula. The essential difficulty is that, in contrast to terized by the fact that Lx(π,s) = Lx(σ (π),s) for the case of Shimura varieties, the moduli stack every place x of F. Shtr is not of finite type, not even at any finite level N. To understand why, recall that a shtuka Theorem 2 (the Ramanujan-Petersson conjec- of rank r is a vector bundle of rank r on X with additional structure (essentially a meromorphic ture). Let π ∈Ar with central character of finite order. Then for every place x of F where π is un- descent datum under Frobenius). Here X is the smooth irreducible projective curve over Fp with ramified, the Hecke eigenvalues z1,...,zr ∈ C are all of absolute value 1. function field F. And, just as the moduli stack of vector bundles of rank r on X is not of finite type, Sht ∅ Sht Theorem 3 (the Deligne conjecture). Let σ ∈G neither are the stacks r, and r,N. r To deal with this difficulty, Lafforgue introduces with determinant character of finite order. Then σ ≤ ≤ the open substacks Sht P and Sht P of shtukas is pure of weight 0, i.e., for any place x of F where r,∅ r,N where the Harder-Narasimhan polygon is bounded σ is unramified, the images of the Frobenius eigen- by P. These substacks are of finite type, and their values z ,...,z under any embedding of Q¯ into 1 n union is the whole space. The trouble is that they C are of absolute value 1. are not stable under the Hecke correspondences. Here Theorems 2 and 3 are consequences of Therefore Lafforgue constructs in the case without ≤P Theorem 1 through P. Deligne’s purity theorem level structure a smooth compactification Shtr,∅ of ≤P and the estimate on Hecke eigenvalues of Jacquet Shtr,∅ with a normal crossing divisor at infinity and and J. Shalika. Theorem 1 itself is proved by in- extends the Hecke correspondences to it by duction on r (Deligne recursion principle). After simple normalization. He then applies the Grothen- what was known before (in addition to the func- dieck-Lefschetz fixed-point formula to these tional equations, essentially the converse theo- correspondences. However, only a part of this for- rems of A. Weil and I. Piatetskii-Shapiro and the mula can be determined explicitly, and therefore FEBRUARY 2003 NOTICES OF THE AMS 213 this seems a pointless exercise. Lafforgue circum- Since 2000 he has been a professor at the Institut vents this problem by isolating the r-essential part des Hautes Études Scientifiques. ≤P of the cohomology of Shtr,∅ and by showing that Further Information the remainder, both the difference between the For excellent overviews of Lafforgue’s proof, see ≤P cohomology of Shtr,∅ and of Shtr,∅ and the coho- Laumon’s Bourbaki seminar No. 873, March 2000, mol ≤P ogy of the boundary of Shtr,∅ , is r-negligible. which also contains an annotated bibliography, Here the work of R. Pink on Deligne’s conjecture and Lafforgue’s notes of a course at the Tata In- on the Grothendieck-Lefschetz formula enters in stitute (http://www.ihes.fr/PREPRINTS/ a decisive way.
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